Abelian variety isogeny class 5.2.ad_e_ag_k_ao over $\F_{2}$

• Label: 5.2.ad_e_ag_k_ao
• Base field: $\F_{2}$
• Dimension: $5$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2}$
Dimension:	$5$
L-polynomial:	$( 1 + x + 2 x^{2} )( 1 - 4 x + 6 x^{2} - 4 x^{3} + 2 x^{4} - 8 x^{5} + 24 x^{6} - 32 x^{7} + 16 x^{8} )$
	$1 - 3 x + 4 x^{2} - 6 x^{3} + 10 x^{4} - 14 x^{5} + 20 x^{6} - 24 x^{7} + 32 x^{8} - 48 x^{9} + 32 x^{10}$
Frobenius angles:	$\pm0.0377785699724$, $\pm0.148391828106$, $\pm0.398391828106$, $\pm0.615026728081$, $\pm0.787778569972$
Angle rank:	$3$ (numerical)
Jacobians:	$1$
Cyclic group of points:	yes

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/4, 1/4, 1/4, 1/4, 3/4, 3/4, 3/4, 3/4, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4$	$776$	$10756$	$1081744$	$20134444$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$4$	$0$	$16$	$20$	$64$	$140$	$304$	$540$	$824$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is not hyperelliptic):

• $x z + y^2 + y z = x^2 + x z + y t + y u + z t + z u + t u = x^2 + x y + x u + y u + z^2 + z t + t^2 + t u + u^2 = 0$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{8}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.b $\times$ 4.2.ae_g_ae_c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.2.b : \(\Q(\sqrt{-7}) \).
• 4.2.ae_g_ae_c : 8.0.18939904.2.

Endomorphism algebra over $\overline{\F}_{2}$

The base change of $A$ to $\F_{2^{8}}$ is 1.256.bf $\times$ 4.256.q_ds_glk_jitk. The endomorphism algebra for each factor is:

• 1.256.bf : \(\Q(\sqrt{-7}) \).
• 4.256.q_ds_glk_jitk : the quaternion algebra over \(\Q(\sqrt{-7 +4 \sqrt{2}})\) with the following ramification data at primes above $2$, and unramified at all archimedean places:

$v$	($ 2 $,\( \pi \))	($ 2 $,\( \pi + 1 \))
$\operatorname{inv}_v$	$1/2$	$1/2$

where $\pi$ is a root of $x^{4} - 2x^{3} + 5x^{2} - 4x + 2$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{2}}$
  The base change of $A$ to $\F_{2^{2}}$ is 1.4.d $\times$ 4.4.ae_i_ai_e. The endomorphism algebra for each factor is:

  • 1.4.d : \(\Q(\sqrt{-7}) \).
  • 4.4.ae_i_ai_e : 8.0.18939904.2.
• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab $\times$ 4.16.a_i_a_q. The endomorphism algebra for each factor is:

  • 1.16.ab : \(\Q(\sqrt{-7}) \).
  • 4.16.a_i_a_q : 8.0.18939904.2.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
5.2.af_m_as_s_as	$2$	(not in LMFDB)
5.2.d_e_g_k_o	$2$	(not in LMFDB)
5.2.f_m_s_s_s	$2$	(not in LMFDB)